PT quantum theory - brown.edu · Latest results on PT quantum theory Carl Bender Washington University Paris 2011

Latest results on

PT quantum theory

Carl Bender

Washington University

Paris 2011

Assumptions of quantum mechanics:

• causality

• locality

• relativistic invariance

• existence of a ground state

• conservation of probability (unitarity)

• positive real energies

• Hermitian Hamiltonian

The point of this talk:

Dirac Hermiticity is too strong an

axiom of quantum mechanics!

• guarantees real energy and conserved probability

• but … is a mathematical axiom and not a

physical axiom of quantum mechanics

H = Hmeans transpose + complex conjugate

32 ixpH

Many people looked at this model...

(1) C.-I Tan, R. Brower, M. Moshe, M. Furman, ...

(Reggeon field theory, the Pomeron, and all that)

(2) J. Cardy, G. Mussardo, M. Fisher, A. Zamolodchikov, ...

(Lee-Yang edge singularity)

Wait a minute…

this Hamiltonian has

PT symmetry!

32 ixpH

P = parity

T = time reversal

Perturbative solution:

Some references …

• CMB and S. Boettcher, Physical Review Letters 80, 5243 (1998)

• CMB, D. Brody, H. Jones, Physical Review Letters 89, 270401 (2002)

• CMB, D. Brody, and H. Jones, Physical Review Letters 93, 251601 (2004)

• CMB, D. Brody, H. Jones, B. Meister, Physical Review Letters 98, 040403 (2007)

• CMB and P. Mannheim, Physical Review Letters 100, 110402 (2008)

• CMB, D. Hook, P. Meisinger, Q. Wang, Physical Review Letters 104, 061601 (2010)

• CMB and S.Klevansky, Physical Review Letters 105, 031602 (2010)

• CMB, Reports on Progress in Physics 70, 947 (2007)

• P. Dorey, C. Dunning, and R. Tateo, Journal of Physics A 34, 5679 (2001)

• P. Dorey, C. Dunning, and R. Tateo, Journal of Physics A 40, R205 (2007)

How to prove that the

eigenvalues are real

The proof is difficult! It uses

techniques from conformal field

theory and statistical mechanics:

(1) Bethe ansatz

(2) Monodromy group

(3) Baxter T-Q relation

(4) Functional Determinants

Other recent PT papers …

• K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, Phyical Review Letters 100, 103904 (2008)

• Z. Musslimani, K. Makris, R. El-Ganainy, and D. Christodoulides, Physical Review Letters 100, 030402 (2008)

• U. Günther and B. Samsonov, Physical Review Letters 101, 230404 (2008)

• E. Graefe, H. Korsch, and A. Niederle, Physical Review Letters 101, 150408 (2008)

• S. Klaiman, U. Günther, and N. Moiseyev, Physical Review Letters 101, 080402 (2008)

• U. Jentschura, A. Surzhykov, and J. Zinn-Justin, Physical Review Letters 102, 011601 (2009)

• A. Mostafazadeh, Physical Review Letters 102, 220402 (2009)

• O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Physical Review Letters 103, 030402 (2009)

• S. Longhi, Physical Review Letters 103, 123601 (2009)

• A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Physical Review Letters 103, 093902 (2009)

• H. Schomerus, Physical Review Letters 104, 233601 (2010)

• S. Longhi, Physical Review Letters 105, 013903 (2010)

• C. West, T. Kottos, T. Prosen, Physical Review Letters 104, 054102 (2010)

• S. Longhi, Physical Review Letters 105, 013903 (2010)

• T. Kottos, Nature Physics 6, 166 (2010)

• C. Ruter, K. Makris, R. El-Ganainy, D. Christodoulides, M. Segev, and D. Kip, Nature Physics 6, 192 (2010)

• Y. D. Chong, L. Ge, and A. D. Stone, Physical Review Letters 106, 093902 (2011)

• Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Physical Review Letters 106, 213901 (2011)

PT BoundaryRegion of unbroken

PT symmetry

Region of broken

PT symmetry

Broken ParroT Unbroken ParroT

Broken PT symmetry in Paris

The PT Boundary is a phase

transition – at the classical level

(Ask me after the talk if you’re interested!)

OK, so the eigenvalues are real …

But is this quantum mechanics??

• Probabilistic interpretation??

• Hilbert space with a positive metric??

• Unitarity??

The Hamiltonian determines its own adjoint

Unitarity

With respect to the CPT adjoint

the theory has UNITARY time

evolution.

Norms are strictly positive!

Probability is conserved!

OK, so we have unitarity…

But is PT quantum mechanics useful??

• Revives quantum theories that were thought

to be dead

• Observed experimentally

Lee Model

The problem with the Lee Model:

“A non-Hermitian Hamiltonian is unacceptable

partly because it may lead to complex energy

eigenvalues, but chiefly because it implies a non-

unitary S matrix, which fails to conserve probability

and makes a hash of the physical interpretation.”

PT quantum mechanics to the rescue…

Meep! Meep!

PT

Gives a fourth-order field equation:

Pais-Uhlenbeck action

The problem: A fourth-order field

equation gives a propagator like

GHOST!

Two possible realizations…

There is another realization as well!

The Hamiltonian is not Dirac Hermitian, but it is

PT symmetric. We can calculate the C operator

exactly. Norm is positive and the spectrum is bounded

below. This suggests how Pauli-Villars works.

No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck

model, CMB and P. Mannheim, Physical Review Letters 100, 110402 (2008)

CMB and P. Mannheim, Physical Review D 78, 025002 (2008)

Laboratory verification using

table-top optics experiments!

• Z. Musslimani, K. Makris, R. El-Ganainy, and D.

Christodoulides, Physical Review Letters 100, 030402 (2008)

• K. Makris, R. El-Ganainy, D. Christodoulides, and Z.

Musslimani, Physical Review Letters 100, 103904 (2008)

• A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-

Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides,

Physical Review Letters 103, 093902 (2009)

• C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides,

M. Segev, and D. Kip, Nature Physics 6, 192 (2010)

Observing PT symmetry using optical wave guides:

The observed PT phase transition

Another experiment...

“Enhanced magnetic resonance signal of spin-polarized Rb

Atoms near surfaces of coated cells”

K. F. Zhao, M. Schaden, and Z. Wu

Physical Review A 81, 042903 (2010)

And another experiment...

Spontaneous PT-symmetry breakdown in superconducting weak

links

N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, V. M. Vinokur(arXiv:1008.3590v2 [cond-mat.supr-con], submitted on 21 Aug 2010 (v1), last revised 1 Sep 2010(v2))

Abstract: We formulate a description of transport in a superconducting weak link in

terms of the non-Hermitian quantum mechanics. We find that the applied electric field

exceeding a certain critical value change the topological structure of the effective non-

Hermitian Hamiltonian of the weak link in the Hilbert space causing the parity

reflection – time reversal symmetry (PT-symmetry) breakdown. We derive the

expression of the critical electric field and show that the PT-symmetry breakdown gives

rise to the switching instability in the current-voltage characteristic of the weak link.

Taking into account superconducting fluctuations we quantitatively describe the

experimentally observed differential resistance of the weak link in the vicinity of the

critical temperature.

And yet another...

Spontaneous Parity--Time Symmetry Breaking and Stability of Solitons

in Bose-Einstein Condensates

Zhenya Yan, Bo Xiong, Wu-Ming Liu

(arXiv:1009.4023v1 [cond-mat.quant-gas], submitted on 21 Sep 2010)

Abstract: We report explicitly a novel family of exact PT-symmetric solitons and

further study their spontaneous PT symmetry breaking, stabilities and collisions in

Bose-Einstein condensates trapped in a PT-symmetric harmonic trap and a Hermite-

Gaussian gain/loss potential. We observe the significant effects of mean-field

interaction by modifying the threshold point of spontaneous PT symmetry breaking

in Bose-Einstein condensates. Our scenario provides a promising approach to

study PT-related universal behaviors in non-Hermitian quantum system based on the

manipulation of gain/loss potential in Bose-Einstein condensates.

Interesting recent developments...(1) K. Jones-Smith and H. Mathur (Case Western): PT-symmetric Dirac

equation and neutrino oscillations

(2) G. „t Hooft: cosmological models

(3) J. Moffat (Perimeter): cosmological constant

(4) M. de Kieviet (Heidelberg): experimental observations of PT-symmetric

quantum brachistochrone

(5) P. Dorey, C. Dunning, R. Tateo: ODE-IM correspondence

(6) D. Masoero (Trieste): cubic PT oscillator and Painleve I; quartic PT

oscillator and Painleve II

(7) S. Longhi (Milan): Bloch waves

(8) Classical PT-symmetric equations: KdV, Camassa-Holm, Sine-Gordon,

Boussinesq, Lotka-Volterra, Euler‟s; complex extension of chaos

(9) Complex quantum mechanics: Complex correspondence principle

(10) A. LeClair (Cornell): Generalization of spin and statistics

(11) H. Schomerous (Lancaster): PT quantum noise

(12) D. Christodoulides (Florida): Random PT dimers

(13) ... And lots more!

OK, but how do we interpret a

non-Hermitian Hamiltonian??

Solve the quantum brachistochrone problem…

Quantum brachistochrone

Constraint:

Hermitian case

becomes:

Minimize t over all positive r

while maintaining constraint

Minimum evolution time:

Looks like uncertainty principle but is merely

rate times time = distance

Non-Hermitian PT-symmetric Hamiltonian

where

The bottom line…

So, what does PT symmetry really mean?

Interpretation…

Finding the optimal PT-symmetric

Hamiltonian amounts to constructing

a wormhole in Hilbert space!

“The shortest path between two

truths in the real domain passes

through the complex domain.”

-- Jacques Hadamard

The Mathematical

Intelligencer 13 (1991)

Thanks for listening!

OK, but what exactly is this

PT phase transition?

Examining the CLASSICAL limit of PT quantum mechanics

provides an intuitive explanation of PT symmetry…

Motion on the real axis

Motion of particles is governed by Newton‟s Law:

F=maIn freshman physics this motion is restricted to the

REAL AXIS.

Harmonic oscillator:

Particle on a spring

Turning point Turning point

Back and forth motion on the real axis:

( = 0)

Hamilton’s equations

Harmonic oscillator:

Turning point Turning point

Motion in the

complex plane:

( = 0)

32 ixpH ( = 1)

e p - 2

Classical orbit that visits three sheets of

the Riemann surface

= – 2 11 sheets

Broken PT symmetry – orbit not closed

e< 0

p